Reflector telescopes gather and shape light differently than refractor scopes, which are built to refract or bend light to form an image. Instead, reflecting telescopes use mirrors to reflect light, which is then focused and observed through an ocular or eyepiece.
Design-wise, a reflecting telescope features a concave mirror (the "primary") at its bottom, which collects the light. The light is then focused at the "focal plane." The "focal length" is the distance from the primary to the focal plane. This design eliminates chromatic aberration, though spherical aberration can still be an issue due to misalignment of mirrors.
The first reflecting telescope is believed to have been built by Niccolo Zucchi, an Italian monk and astronomer who used his invention to lay claim to a number of discoveries, including being the first to identify belts of the planet Jupiter in 1630; an examination of spots on Mars a decade later; and in 1652, the idea that phosphors didn't store light, but rather, generated it. His telescope design would prove the inspiration for James Gregory and Isaac Newton.
Of the reflector designs, the two main ones are Cassegrain and Newtonian. In the Cassegrain model, there is a primary concave mirror (parabolic) which collects light, then a secondary convex mirror (hyperbolic) that reflects that light back through the primary. Both mirrors are aligned about the optical axis (the line which the light path passes). By "folding" the optics, the Cassegrain telescope can be a very compact design, of which there are also variants: Ritchey-Chretien, Dall-Kirkham, Schiefspiegler, and Gregorian (by James Gregory).
As for the Newtonian refractor telescope, Isaac Newton took the design by Zucchi and instead of a secondary mirror as is found in the Cassegrain model, a flat mirror is placed at a diagonal. This reflects the light at an angle into an eyepiece that is located on the side of the tube. The observer is thus positioned at the top of the scope, looking into the ocular which reveals the focused image.
As with refractors, there are advantages and disadvantages to the reflecting telescope, from the (positive) elimination of chromatic aberration to the (negative) potential problems of misalignment (e.g. the Hubble Telescope's early years). These are some considerations that need to be asked before deciding on which type of telescope is best, as a starter, a secondary, or upgrade. It should be said that the Newtonian refractor is a popular choice for first-time telescope builders.
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